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Theorem smgrpisass 21000
Description: A semi-group is associative. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
Assertion
Ref Expression
smgrpisass  |-  ( G  e.  SemiGrp  ->  G  e.  Ass )

Proof of Theorem smgrpisass
StepHypRef Expression
1 elin 3358 . . 3  |-  ( G  e.  ( Magma  i^i  Ass ) 
<->  ( G  e.  Magma  /\  G  e.  Ass )
)
21simprbi 450 . 2  |-  ( G  e.  ( Magma  i^i  Ass )  ->  G  e.  Ass )
3 df-sgr 20998 . 2  |-  SemiGrp  =  (
Magma  i^i  Ass )
42, 3eleq2s 2375 1  |-  ( G  e.  SemiGrp  ->  G  e.  Ass )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684    i^i cin 3151   Asscass 20979   Magmacmagm 20985   SemiGrpcsem 20997
This theorem is referenced by:  mndoisass  24768
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-in 3159  df-sgr 20998
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